quantum condensed matter physics: organic insulators and...

Quantum condensed matter physics:organic insulators

and ultracold atoms

HARVARDsachdev.physics.harvard.eduWednesday, March 2, 2011

Outline

1. Organic insulators: antiferromagnets on the triangular lattice

2. Ultracold atoms: bosons in tilted Mott insulators

Wednesday, March 2, 2011

Half-filled band à Mott insulator with spin S = 1/2Triangular lattice of [Pd(dmit)2]2

à frustrated quantum spin system

X[Pd(dmit)2]2 Pd SC

X Pd(dmit)2

t’tt


H =�

�ij�

Jij�Si · �Sj + . . .

H = J

�

�ij�

�Si · �Sj ; �Si ⇒ spin operator with S = 1/2


Found in -(ET)2Cu[N(CN)2]Clκ

Anisotropic triangular lattice antiferromagnet

Classical ground state for small J’/J



Classical ground state for large J’/JFound in Cs2CuCl4


Valence bond solid


N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

=


M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006)Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007)

Observation of a valence bond solid (VBS) in ETMe3P[Pd(dmit)2]2

Spin gap ~ 40 K J ~ 250 K

X-ray scattering


Magnetic Criticality

t’/t

T N (K

)

Magnetic order

Quantum critical

Spin gap

Me4P

Me4As

EtMe3As

Et2Me2As Me4Sb

Et2Me2P

EtMe3Sb

EtMe3Pt’/t = 1.05

Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)

X[Pd(dmit)2]2Et2Me2Sb (CO)

VBS order


=

Triangular lattice antiferromagnet

Z2 spin liquid

P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)


=

Triangular lattice antiferromagnet

Z2 spin liquid

P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

Candidate for aZ2 spin liquid:

κ-(ET)2Cu2(CN)3


=

-1

Excitations of the Z2 Spin liquid

A vison



=

-1

-1


A vison



• A characteristic property of a Z2 spin liquidis the presence of a spinon pair condensate

• A vison is an Abrikosov vortex in the paircondensate of spinons

• Visons are are the dark matter of spin liq-uids: they likely carry most of the energy,but are very hard to detect because they donot carry charge or spin.


A vison



Effective description of Z2 spin liquids, their visons and valence bond solids

Quantum dimer model:

D. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988)R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001)

H =V

��

�� + V

��

��

− J

��

�� − J

��

��

Hilbert space - set of dimer coverings of triangular/square lattice


Outline

1. Organic insulators: antiferromagnets on the triangular lattice

2. Ultracold atoms: bosons in tilted Mott insulators


SusannePielawa

TakuyaKitagawa

ErezBerg

S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, 075128 (2002)S. Pielawa, T. Kitagawa, E. Berg, S. Sachdev, arXiv:1101.2897


M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Superfluid-insulator transition of 87Rb atoms in a magnetic trap and an optical lattice potential


M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Mott insulator of 87Rb atoms in a magnetic trap and an optical lattice potential


Applying an “electric” field to the Mott insulator


Why is there a peak (and

not a threshold)

when E = U ?


Resonant transition when E≈U


Virtual state


Resonant transition when E≈U


Hamiltonian of resonant subspace


H = −√

2t

�

i

�d†i + di

�+ ∆

�

i

d†i di

∆ = U − E


dipole


H = −√

2t

�

i

�d†i + di

�+ ∆

�

i

d†i di

∆ = U − E


d†i di ≤ 1 d†i did†i+1di+1 = 0

no neighboring dipoles:max one dipole per site:Constraints:

dipole


H = −√

2t

�

i

�d†i + di

�+ ∆

�

i

d†i di

∆ = U − E


d†i di ≤ 1 d†i did†i+1di+1 = 0

no neighboring dipoles:max one dipole per site:Constraints:

dipole

Strong off-site quantum correlations


Phase diagram of dipole model

(E-U)/t

S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, 075128 (2002)


(E-U)/tIsing quantum

phase transition

S. Sachdev, K. Sengupta, and S.M. Girvin, Phys. Rev. B 66, 075128 (2002)

Phase diagram of dipole model


e

Tilting a decorated square lattice

SusannePielawa


e



e


Strong tilt: maximize sites with 2 bosons


e


Maximum number of 2’s


e


Can also get some 3’s from neighboring 2’s.


e


No more 3’s are possible, but some 2’s are left over


e


Start againWednesday, March 2, 2011

e


Another maximal set of 2’sWednesday, March 2, 2011

e


Maximum number of 3’s with no 2’s left over


e


Configurations map onto dimer coverings of the square lattice !

SusannePielawa


e


Go backwards around a plaquetteWednesday, March 2, 2011

e


Then create a different set of 3’sWednesday, March 2, 2011

e


A different dimer coveringWednesday, March 2, 2011

e


SusannePielawa

Dimers can resonate around a plaquette


e


Dimers can resonate around a plaquette

SusannePielawa


e


SusannePielawa

Strong tilt: effective quantum dimer model


Many common issues on many body quantum correlations in condensed matter and ultracold atoms

Tilting Mott insulators can generate many interesting states with non-trivial quantum entanglement

Conclusions


quantum condensed matter physics: organic insulators and...

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